Let $H$ be a Hilbert space and $B(H)$ the algebra of all bounded linear operators on $H$. In this paper we shall show that if $A \in B(H)$ is a nonzero closed range operator, then the injective norm $\Vert A^{*}\otimes A^{+}+A^{+}\otimes A^{*}\Vert_{\lambda}$ attains its minimal value 2 if and only if $A/\Vert A\Vert$ is a partial isometry. Also we shall give some characterizations of partial isometries and normal partial isometries in terms of norm equalities for operators. These characterizations extend previous ones obtained by A. Seddik in [On the injective norm and characterization of some subclasses of normal operators by inequalities or equalities, J. Math. Anal. Appl. 351 (2009), 277--284], and by M. Khosravi in [A characterization of the class of partial isometries, Linear Algebra Appl. 437 (2012)].